Given two right triangles, $△\mathrm{PQR}$ with sides PQ = $3$, QR = $4$, and RP = $5$, and $△\mathrm{STU}$ with sides ST = $5$, TU = $12$, and US = $13$, which angle has a sine ratio of $\frac{3}{5}$?

• A. Angle Q in $△\mathrm{PQR}$
• B. Angle S in $△\mathrm{STU}$
• C. Angle T in $△\mathrm{STU}$
• D. Angle P in $△\mathrm{PQR}$

Answer: (D)

Verified step by step guidance

1. Identify the sides of triangle \( \triangle PQR \) with lengths \( PQ = 3 \), \( QR = 4 \), and \( RP = 5 \). Since \( RP = 5 \) is the longest side, it is the hypotenuse of this right triangle.
2. Recall that the sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse: \[ \sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}} \]
3. For each angle in \( \triangle PQR \), determine which side is opposite and calculate the sine ratio using the side lengths. For example, for angle \( P \), the side opposite is \( QR = 4 \), and the hypotenuse is \( RP = 5 \), so \( \sin(P) = \frac{4}{5} \). Repeat this for angles \( Q \) and \( R \).
4. Repeat the same process for \( \triangle STU \) with sides \( ST = 5 \), \( TU = 12 \), and \( US = 13 \), where \( US = 13 \) is the hypotenuse. Calculate the sine ratios for angles \( S \), \( T \), and \( U \) by identifying the opposite sides and dividing by the hypotenuse.
5. Compare the sine ratios you calculated for all angles to the given ratio \( \frac{3}{5} \). The angle whose sine ratio matches \( \frac{3}{5} \) is the correct answer.